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mac1140_lecture23_2

# mac1140_lecture23_2 - L23 5.3 Logarithmic Functions The...

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200 L23 §5.3 Logarithmic Functions The exponential function ( ) x f x a = ( 0, 1) a a > is a 1-1 function and, therefore, it has the inverse 1 ( ) f x . We denote the inverse log a x and read “logarithm to the base a of x ”. Sketch the graphs of 2 x y = and 2 log y x = . Sketch the graphs of 1 2 x y = and 1 2 log y x = 201 We use what we know about x y a = and inverses to get the properties of log a y x = . ( ) x f x a = 1 ( ) log a f x x = Points (0,1) and (1, ) a are on the graph Points ( , ) and ( , ) are on the graph 0 y = is HA Domain: ( , ) −∞ +∞ Domain: ( , ) Range: (0, ) +∞ Range: ( , ) log x a a x = for any real x log a x a x = for 0 x > **Important** Since inverse functions undo each other with respect to composition, the following identities hold: log for all real x a a x x = log for 0 a x a x x = > . We use these identities for solving exponential and logarithmic equations and inequalities.

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